I need help creating and understanding how to write a proof about the opposite o

Question

I need help creating and understanding how to write a proof about the opposite of the below. I need to prove how to take a octal number and turn it into a binary number. Below is the given example of the reverse (binary to octal). I understand that (base2)3 = base8. But how to I show mathatically with a formula that 78 = 111?

We have learned that we can block-convert a binary number to octal by grouping the binary number into blocks of 3 digits (going from right to left, perhaps padding the binary number with one or two leading zeros to complete the leftmost block of three) and converting each 3 digit binary number into one octal digit. The following is a proof that this procedure works as advertised. Suppose n is a nonnegative integer and its binary expansion is given by where each dk E {0,1} and m is a nonnegative integer. We can assume without loss of generality that the number of terms in this sum is a multiple of 3,i.e m 1 3q for some natural number q We now group the sum into blocks of 3 terms each, as follows q-1 2 we now set oi = d3it 2 We now set o -d3i + 2d3 d3i+1 + 4d3i+2 for all i and get Since each dk is 0 or 1, the o satisfy 0 o S7, i.e., there are octal digits. We have found the octal expansion of n, and it is obtained by block-converting three binary digits at a time too octal, from right to left.

Explanation / Answer

Here i am showing the another example which shows that how to convert an octal number into binary number ..i hope it would help

Binary to Octal Conversion :-

As the binary numbers are comprised of only 0 and 1 we have to first divide the binary number into group of 3 digits starting from the right most side, and balancing the number of digits by putting appropriate number of zeroes. Now we have certain numbers of group of binary numbers, then we have to convert those groups into decimal numbers and write in the same order in which they used to be. Now this is the octal equivalent of that binary number. If we consider an example, this will be very easy to understand.
Let us take a binary number say 1011010012
Now dividing it into group of three digits we will find the following pattern
101|101|001
Now writing the equivalent decimal number of each group we get
5 | 5 | 1
So the equivalent octal number is 5518

Octal to Binary Conversion :-

Now if we just imagine the reverse of the above explained operation we will be able to find out the method for converting octal number into binary number ourselves. In this case each of the digits of the octal number is converted into its equivalent binary number and they are merged into the same order they were when they were as octal numbers, the leftmost zeroes are omitted from the number and we get the equivalent binary number. An example is given hereby to explain the method easily.
Let us take a number 348
Now the binary equivalent of 3 is 011 and that of 4 is 100.
So, the desired binary equivalent of the octal number is 111002.

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